How to show the completeness of the following subset of $\ell^2$?
Solution 1:
Possibly you are not asking the question you'd really want to ask.
First, no, the closure of that set is the whole $\ell^2$, because it includes all finite linear combinations of standard basis elements.
But it might be (as in some real-life applications, e.g., Sobolev spaces) that you want to define a new norm by (the square root of) the expression that you wrote. This gives a new inner-product-space structure, and (as with $\ell^2$ itself) gives a complete space.